We introduce Clifford Group Equivariant Neural Networks: a novel approach for constructing $\mathrm{O}(n)$- and $\mathrm{E}(n)$-equivariant models. We identify and study the $\textit{Clifford group}$, a subgroup inside the Clifford algebra whose definition we adjust to achieve several favorable properties. Primarily, the group's action forms an orthogonal automorphism that extends beyond the typical vector space to the entire Clifford algebra while respecting the multivector grading. This leads to several non-equivalent subrepresentations corresponding to the multivector decomposition. Furthermore, we prove that the action respects not just the vector space structure of the Clifford algebra but also its multiplicative structure, i.e., the geometric product. These findings imply that every polynomial in multivectors, An advantage worth mentioning is that we obtain expressive layers that can elegantly generalize to inner-product spaces of any dimension. We demonstrate, notably from a single core implementation, state-of-the-art performance on several distinct tasks, including a three-dimensional $n$-body experiment, a four-dimensional Lorentz-equivariant high-energy physics experiment, and a five-dimensional convex hull experiment.

翻译：我们提出Clifford群等变神经网络：一种构建$\mathrm{O}(n)$与$\mathrm{E}(n)$等变模型的新方法。我们识别并研究了$\textit{Clifford群}$，即Clifford代数的一个子群，通过调整其定义以实现若干有利性质。该群的作用形成一种正交自同构，不仅作用于标准向量空间，更扩展至整个Clifford代数，同时保持多重向量分级结构。由此导出对应多重向量分解的多个非等价子表示。进一步证明，该作用不仅保持Clifford代数的向量空间结构，还保持其乘法结构（即几何积）。这些结论表明，多重向量的任意多项式，若包含Clifford群作用下保持的多项式映射，均自动满足等变性特征。因此，我们构建的等变层可高效处理多重向量表示。该方法统一了诸多现有等变神经网络，包括$\mathrm{O}(n)$等变图神经网络、$\mathrm{E}(n)$等变消息传递网络、群卷积网络、等变Transformer及基于扩张群的模型。值得提及的优势是，我们获得了可优雅泛化至任意维内积空间的表达性层。我们证明，通过单一核心实现，该方法在多项不同任务上达到最先进性能，包括三维$n$体实验、四维Lorentz等变高能物理实验及五维凸包实验。